staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Koopa Tak-Lun Koo; William Stein; Gabor Wiese
On the generation of the coefficient field of a newform by a single Hecke eigenvalue
Journal de théorie des nombres de Bordeaux, 20 no. 2 (2008), p. 373-384, doi: 10.5802/jtnb.633
Article PDF | Reviews MR 2477510 | Zbl 1171.11027

Résumé - Abstract

Let $f$ be a non-CM newform of weight $k \ge 2$. Let $L$ be a subfield of the coefficient field of $f$. We completely settle the question of the density of the set of primes $p$ such that the $p$-th coefficient of $f$ generates the field $L$. This density is determined by the inner twists of $f$. As a particular case, we obtain that in the absence of nontrivial inner twists, the density is $1$ for $L$ equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions for further investigation.

Bibliography

[1] S. Baba, R. Murty, Irreducibility of Hecke Polynomials. Math. Research Letters 10 (2003), no. 5-6, 709–715.  MR 2024727 |  Zbl pre02064726
[2] D. W. Farmer, K. James, The irreducibility of some level 1 Hecke polynomials. Math. Comp. 71 (2002), no. 239, 1263–1270.  MR 1898755 |  Zbl 0995.11032
[3] W. Fulton, J. Harris, Representation Theory, A First Course. Springer, 1991.  MR 1153249 |  Zbl 0744.22001
[4] J. Kevin, K. Ono, A note on the Irreducibility of Hecke Polynomials. J. Number Theory 73 (1998), 527–532.  MR 1658012 |  Zbl 0931.11011
[5] K. A. Ribet, Twists of modular forms and endomorphisms of abelian varieties. Math. Ann. 253 (1980), no. 1, 43–62.  MR 594532 |  Zbl 0421.14008
[6] K. A. Ribet, On l-adic representations attached to modular forms. II. Glasgow Math. J. 27 (1985), 185–194.  MR 819838 |  Zbl 0596.10027
[7] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev. Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401. Numdam |  MR 644559 |  Zbl 0496.12011
[8] W. Stein, Sage Mathematics Software (Version 3.0). The SAGE Group, 2008, http://www.sagemath.org.